SHORT ANSWER TYPE QUESTIONS
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Find the equations of the tangents to 9x² + 16y2 = 144, which make equal intercepts on the coordinate axes
Answers
EXPLANATION.
Equation of tangents,
⇒ 9x² + 16y² = 144.
Make an equal intercept on the Co-ordinates axis.
As we know that,
General equation of an ellipse.
⇒ x²/a² + y²/b² = 1. (a > 1).
Compare the equation, we get.
⇒ 9x²/144 + 16y²/144 = 1.
⇒ x²/16 + y²/9 = 1.
As we know that,
Equation of tangent.
Slope form :
⇒ y = mx ± √a²m² + b².
As we know that,
If equation makes an equal intercept on the Co-ordinates axes then,
Their Slope = -1 = m.
Put the value of m = -1 in equation, we get.
⇒ y = mx ± √a²m² + b².
⇒ y = (-1)x ± √16(-1)² + (9).
⇒ y = - x ± √16 + 9.
⇒ y = - x ± √25.
⇒ y = - x ± 5.
⇒ y + x = ± 5.
MORE INFORMATION.
Conditions of tangency.
(1) = The line y = mx + c touches the ellipse x²/a² + y²/b² = 1,
if, c = ± √a²m² + b².
(2) = The equation of the chord of the ellipse x²/a² + y²/b² = 1, whose mid point be (x₁, y₁) is T = S₁.
(3) = The equation of the chord of contact is.
xx₁/a² + yy₁/b² = 1 Or T = 0 at (x₁, y₁).
(4) = Pair of tangents : SS₁ = T².