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Condition for a line to touch an ellipse

This type of question, asked 4 times: 2021 Paper 17, 2021 Paper 18, 4 Apr 2025, 21 Jan 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    84
    Let EE be an ellipse whose axes are parallel to the co-ordinate's axes, having its center at (3,−4)(3, - 4), one focus at (4,−4)(4, - 4) and one vertex at (5,−4)(5, - 4). If mx−y=4, m>0mx - y = 4,\text{ }m > 0 is a tangent to the ellipse EE, then the value of 5 m25{\text{ }m}^{2} is equal to

  2. Question 2

    2021 Paper 18

    63
    On the ellipse x28+y24=1\frac{x^{2}}{8}+\frac{y^{2}}{4}= 1 let PP be a point in the second quadrant such that the tangent at PP to the ellipse is perpendicular to the line x+2y=0x + 2y = 0. Let SS and S′S^{'} be the foci of the ellipse and ee be its eccentricity. If A is the area of the triangle SPS' then, the value of (5−e2)\left( 5 -e^{2} \right). AA is:

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  3. Question 3

    4 Apr 2025

    139
    Let for two distinct values of pp the lines y=x+py=x+p touch the ellipse E:x242+y232=1E:\frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{2}}= 1 at the points A and B . Let the line y=xy = x intersect E at the points C and D . Then the area of the quadrilateral ABCD is equal to.

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  4. Question 4

    21 Jan 2026 S2

    54
    If the line αx+4y=7\alpha x + 4y =\sqrt{7}, where α∈R\alpha \in R, touches the ellipse 3x2+4y2=13x^{2}+ 4y^{2}= 1 at the point PP in the first quadrant, then one of the focal distances of P is :

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