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MHT-CET Maths · Conic Sections

The Parabola — Standard Forms and Tangents

The four standard parabolas, their focus, directrix and latus rectum, a shifted parabola found by completing the square, and the tangent y = mx + a/m with what it does for pairs of tangents and common tangents.

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Why this matters

7 PYQs, two HARD. Two read a parabola's parts — a directrix after completing the square, the triangle on the latus rectum. Five are tangents: the condition c = a/m, a tangent parallel to a line, the angle between the two tangents from a point, a common tangent to two parabolas, and the angle at which two parabolas cross. Two cards.

Concept 1 of 2: Standard Forms, Focus, Directrix and Latus Rectum

A parabola is the set of points as far from a fixed point (the focus) as from a fixed line (the directrix). In y² = 4ax the focus is a to the right of the vertex and the directrix a to the left. The latus rectum is the chord through the focus perpendicular to the axis; its length is 4a. A parabola whose vertex is not at the origin is brought to a standard form by completing the square.

Definition

  • y2=4axy^2 = 4ax: focus (a,0)(a, 0), directrix x=−ax = -a, latus rectum ends (a,±2a)(a, \pm 2a).
  • y2=−4axy^2 = -4ax opens left; x2=4ayx^2 = 4ay opens up, focus (0,a)(0, a), directrix y=−ay = -a; x2=−4ayx^2 = -4ay opens down.
  • Latus rectum length =4a= 4a.
  • Shifted: (y−k)2=4a(x−h)(y - k)^2 = 4a(x - h) has vertex (h,k)(h, k), focus (h+a,k)(h + a, k), directrix x=h−ax = h - a.
  • Complete the square in the squared variable first; the coefficient of the other variable must then be written as ±4a(…)\pm 4a(\ldots).

Parabola y² = 4ax

y2=4ax:S(a,0),x=−a,LR=4ay^2 = 4ax: \quad S(a, 0), \quad x = -a, \quad LR = 4a

Worked example

Find the focus and directrix of y2−6y−8x+17=0y^2 - 6y - 8x + 17 = 0.
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 26 April Shift I · Q111Moderate

Example 1 · Conic Sections · Parabola

The equation of the directrix of the parabola y2+4y+4x+2=0y^{2}+ 4y+ 4x+ 2 = 0 is

Putting the directrix on the same side as the focus

The directrix is a from the vertex on the side AWAY from the focus. (y + 2)² = −4(x − ½) opens left, so the focus is left of x = ½ and the directrix is to the right, at x = 3/2.

Reading 4a as a

In x² = 20y, 4a = 20 and a = 5. The latus rectum is 20 long and lies 5 from the vertex.

Forgetting the sign when completing the square

y² + 4y + 4x + 2 = 0 gives (y + 2)² = −4x + 2 = −4(x − ½). The x-coefficient is negative, so the parabola opens left.
Drill 1 more on standard forms, focus, directrix and latus rectum

Concept 2 of 2: Tangents: y = mx + a/m, Pairs of Tangents and Angles Between Curves

A line meets y² = 4ax in two points, one point, or none. It touches when the quadratic for the meeting points has equal roots, which gives c = a/m. From a point outside, that condition becomes a quadratic in m with two roots — the two tangents. A common tangent to two curves satisfies both conditions at once. The angle between two curves at a point is the angle between their tangents there.

Definition

  • y=mx+cy = mx + c touches y2=4axy^2 = 4ax iff c=amc = \dfrac{a}{m}; it touches x2=4ayx^2 = 4ay iff c=−am2c = -am^2.
  • Tangents from (h,k)(h, k): k=mh+amk = mh + \dfrac{a}{m}, i.e. hm2−km+a=0hm^2 - km + a = 0. Its roots are the two slopes; tan⁡θ=∣m1−m21+m1m2∣\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|.
  • Common tangent to y2=4axy^2 = 4ax and x2=4byx^2 = 4by: m3=−abm^3 = -\dfrac{a}{b}, then c=amc = \dfrac{a}{m}.
  • Angle between curves at a common point: slopes by implicit differentiation, then the same tan θ formula; m1m2=−1m_1m_2 = -1 means they cut at right angles.

Tangency condition

y=mx+c touches y2=4ax  ⟺  c=amy = mx + c \text{ touches } y^2 = 4ax \iff c = \frac{a}{m}

Worked example

Find the tangent to y2=12xy^2 = 12x that is parallel to y=3x+5y = 3x + 5.
Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 26 April Shift I · Q125Moderate

Example 2 · Conic Sections · Parabola

The angle between the tangents drawn from the point (1,4)(1,4) to the parabola y2=4xy^{2}= 4x, is

Using c = a/m for an upward parabola

c = a/m is for y² = 4ax. For x² = 4ay the condition is c = −am². A common-tangent question needs both, one for each curve.

Forgetting the modulus in the angle

The angle between two lines is acute, so take |m₁ − m₂|/|1 + m₁m₂|. From Vieta, |m₁ − m₂| = √((m₁ + m₂)² − 4m₁m₂).

Differentiating the wrong variable

For x² + 4(y − 3) = 0, differentiate with respect to x: 2x + 4y′ = 0, so y′ = −x/2. At (2, 2) this is −1, not 1.
Drill 4 more on tangents: y = mx + a/m, pairs of tangents and angles between curves

Summary — formulas & gotchas at a glance

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Formulas (2)

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