Parabola
General
Grade 11
Question:
Consider the graphs of y = Ax<span class="math-inline">^2</span> and y<span class="math-inline">^2</span> + 3 = x<span class="math-inline">^2</span> + 4y, where A is a positive constant and x,y ∈ R. Number of points in which the two graphs intersect, is-
exactly 4
exactly 2
at least 2 but the number of points varies for different positive values of A.
zero for atleast one positive A.
Step-by-Step Solution
Key Concept: General
Given ellipse are <br> x²/4 + y²/1 = 1 <br> and, <br> x²/6 + y²/3 = 1 <br> any tangent to (i) is x cos θ/2 + y sin θ/1 = 1 <br> It cuts (ii) at P and Q, and suppose tangent at P and Q meet at (h, k) Then equation <br> of chord of contact of (h, k) with respect to ellipse (ii) is hx/6 + ky/3 = 1 <br> comparing (iii) and (iv), we get cos θ/h/3 = sin θ/k/3 = 1 <br> ⇒ cos θ = h/3 and sin θ = k/3 ⇒ h² + k² = 9 <br> locus of the point (h, k) is x² + y² = 9 ⇒ x² + y² = 6 + 3 = a² + b² <br> i.e. director circle of second ellipse. Hence the tangents are at right angles.
Correct Answer: 1